Triple Integral
Author: Joseph Lo
Date: 2 April 2013

We consider the triple integral \[\iiint_E f(x,y,z) \,dV\] where \(E\) is the solid bounded by \(z = 1-x^2\), \(z = 1-y\), \(y = 1\) and \(x = 0\). There are six ways to write this triple integral as an interated integral in Cartesian coordinates: \[\int_0^1 \int_{1-z}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dy \,dz\] \[\int_0^1 \int_{1-y}^1 \int_0^{\sqrt{1-z}} f(x,y,z) \,dx \,dz \,dy\] \[\int_0^1 \int_0^{\sqrt{1-z}} \int_{1-z}^1 f(x,y,z) \,dy \,dx \,dz\] \[\int_0^1 \int_0^{1-x^2} \int_{1-z}^1 f(x,y,z) \,dy \,dz \,dx\] \[\int_0^1 \int_0^{\sqrt{y}} \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dx \,dy\]  \[\int_0^1 \int_{x^2}^1 \int_{1-y}^{1-x^2} f(x,y,z) \,dz \,dy \,dx\]

domain

w.r.t. x first

w.r.t. y first

w.r.t. z first

then y, then x

then x, then y

then x, then z

then z, then x

then y, then z

then z, then y

z = 1 - x²

z = 1 - y

y = 1

x = 0

Surfaces

Integrate

x = 0

x

y

y

z

domaincheck = true
xcheck = false
ycheck = false
zcheck = true
C22 = (0, 1)
A1 = 0
B1: A1^(2)
C1: (A1, B1)
A2: A1 + 0.05
B2: A2^(2)
C2: (A2, B2)
A3: A2 + 0.0500000000000001
B3: A3^(2)
C3: (A3, B3)
A4: A3 + 0.0500000000000001
B4: A4^(2)
C4: (A4, B4)
A5: A4 + 0.0500000000000001
B5: A5^(2)
C5: (A5, B5)
A6: A5 + 0.0500000000000001
B6: A6^(2)
C6: (A6, B6)
A7: A6 + 0.0500000000000001
B7: A7^(2)
C7: (A7, B7)
A8: A7 + 0.0500000000000001
B8: A8^(2)
C8: (A8, B8)
A9: A8 + 0.0500000000000001
B9: A9^(2)
C9: (A9, B9)
A10: A9 + 0.0500000000000001
B10: A10^(2)
C10: (A10, B10)
A11: A10 + 0.0500000000000001
B11: A11^(2)
C11: (A11, B11)
A12: A11 + 0.0500000000000001
B12: A12^(2)
C12: (A12, B12)
A13: A12 + 0.0500000000000001
B13: A13^(2)
C13: (A13, B13)
A14: A13 + 0.0500000000000001
B14: A14^(2)
C14: (A14, B14)
A15: A14 + 0.0500000000000001
B15: A15^(2)
C15: (A15, B15)
A16: A15 + 0.0500000000000001
B16: A16^(2)
C16: (A16, B16)
A17: A16 + 0.0500000000000001
B17: A17^(2)
C17: (A17, B17)
A18: A17 + 0.0500000000000001
B18: A18^(2)
C18: (A18, B18)
A19: A18 + 0.0500000000000001
B19: A19^(2)
C19: (A19, B19)
A20: A19 + 0.0500000000000001
B20: A20^(2)
C20: (A20, B20)
A21: A20 + 0.0500000000000002
B21: A21^(2)
C21: (A21, B21)
S1 = Surface(s, s^(2) + (t * (1 - s^(2))), 1 - s^(2), s, 0, 1, t, 0, 1)
S2 = Surface((s * sqrt(t)), t, 1 - t, s, 0, 1, t, 0, 1)
S3 = Surface(s, 1, (t * (1 - s^(2))), s, 0, 1, t, 0, 1)
S4 = Surface(0, s, 1 - s + (t * s), s, 0, 1, t, 0, 1)
Imagexy = Surface(s, s^(2) + (t * (1 - s^(2))), 0, s, 0, 1, t, 0, 1)
Imagexz = Surface(s, 0, (t * (1 - s^(2))), s, 0, 1, t, 0, 1)
Imageyz = Surface(0, s, 1 - s + (t * s), s, 0, 1, t, 0, 1)
D1: (A1, 1 - B1)
D2: (A2, 1 - B2)
D3: (A3, 1 - B3)
D4: (A4, 1 - B4)
D5: (A5, 1 - B5)
D6: (A6, 1 - B6)
D7: (A7, 1 - B7)
D8: (A8, 1 - B8)
D9: (A9, 1 - B9)
D10: (A10, 1 - B10)
D11: (A11, 1 - B11)
D12: (A12, 1 - B12)
D13: (A13, 1 - B13)
D14: (A14, 1 - B14)
D15: (A15, 1 - B15)
D16: (A16, 1 - B16)
D17: (A17, 1 - B17)
D18: (A18, 1 - B18)
D19: (A19, 1 - B19)
D20: (A20, 1 - B20)
D21: (A21, 1 - B21)
D22 = (0, 0)
Cxy1 = CurveCartesian(t, t^(2), t, 0, 1)
Cxy2 = CurveCartesian(t, 1, t, 0, 1)
Cxy3 = CurveCartesian(0, t, t, 0, 1)
Domainxz = Polygon(D1:D22)
Cxz1 = CurveCartesian(t, 1 - t^(2), t, 0, 1)
Cxz2 = CurveCartesian(t, 0, t, 0, 1)
Cxz3 = CurveCartesian(0, t, t, 0, 1)
Domainxy = Polygon(C1:C22)
showlimit: zcheck ∨ ycheck ∨ xcheck ∨ (¬domaincheck)
Domainyz, Cyz1, Cyz2, Cyz3 = Polygon((1, 0), (0, 1), (1, 1))
P1 = (0, 0, 1)
P2 = (0, 1, 1)
P3 = (0, 1, 0)
P4 = (1, 1, 0)
L12 = Segment(P1, P2)
L23 = Segment(P2, P3)
L34 = Segment(P3, P4)
Py1 = (0.4, 0.5, 0.5)
Py2 = (0.4, 1, 0.5)
Ay = Vector(Py1, Py2)
Pz1 = (0.4, 0.5, 0.5)
Pz2 = (0.4, 0.5, 0.84)
Az = Vector(Pz1, Pz2)
Px1 = (0, 0.7, 0.7)
Px2 = (0.5477225575051661, 0.7, 0.7)
Ax = Vector(Px1, Px2)
zyxcheck = true
Pzyx1 = (0, 0.5)
Azyx = Vector((0.4, 0.16), (0.4, 1))
zxycheck = false
Azxy = Vector((0, 0.6), (0.7745966692414834, 0.6))
Ayzx = Vector((0.4, 0), (0.4, 0.84))
Ayxz = Vector((0, 0.4), (0.7745966692414834, 0.4))
yxzcheck = false
yzxcheck = false
Axyz = Vector((0.4, 0.6), (1, 0.6))
Axzy = Vector((0.6, 0.4), (0.6, 1))
xyzcheck = false
xzycheck = false
Sf1 = Surface(s, t, 1 - s^(2), s, -2, 2, t, -2, 2)
Sf2 = Surface(s, t, 1 - t, s, -2, 2, t, -2, 2)
Sf3 = Surface(s, 1, t, s, -2, 2, t, -2, 2)
Sf4 = Surface(0, s, t, s, -2, 2, t, -2, 2)
Sf1check = false
Sf2check = false
Sf3check = false
Sf4check = false
Ptx1 = (-0.2, 0.7, 0.7)
Axesx = Vector((2, 0.5), (2, 0.75))
Axesy = Vector((2, 0.5), (2.25, 0.5))
Axesty = (2, 0.85)
Axestx = (2.35, 0.5)
temp = 1 (slider from 1 to 2, step 2)